{
  "id": "2203.12250",
  "version": "v1",
  "published": "2022-03-23T07:49:01.000Z",
  "updated": "2022-03-23T07:49:01.000Z",
  "title": "Local Statistics of Random Permutations from Free Products",
  "authors": [
    "Doron Puder",
    "Tomer Zimhoni"
  ],
  "comment": "38 pages, 3 figures",
  "categories": [
    "math.GR",
    "math.CO",
    "math.PR"
  ],
  "abstract": "Let $\\Gamma=G_1*\\ldots*G_k$ be a free product of groups where each of $G_1,\\ldots,G_k$ is either finite, finitely generated free, or an orientable hyperbolic surface group. For a fixed element $\\gamma\\in\\Gamma$, a $\\gamma$-random permutation in the symmetric group $S_N$ is the image of $\\gamma$ through a uniformly random homomorphism $\\Gamma\\to S_{N}$. In this paper we study local statistics of $\\gamma$-random permutations and their asymptotics as $N$ grows. We first consider $\\mathbb{E}\\left[\\mathrm{fix}_{\\gamma}\\left(N\\right)\\right]$, the expected number of fixed points in a $\\gamma$-random permutation in $S_{N}$. We show that unless $\\gamma$ has finite order, the limit of $\\mathbb{E}\\left[\\mathrm{fix}_{\\gamma}\\left(N\\right)\\right]$ as $N\\to\\infty$ is an integer, and is equal to the number of subgroups $H\\le\\Gamma$ containing $\\gamma$ such that $H\\cong\\mathbb{Z}$ or $H\\cong C_{2}*C_{2}$. Equivalently, this is the number of subgroups $H\\le\\Gamma$ containing $\\gamma$ and having (rational) Euler characteristic zero. We also prove there is an asymptotic expansion for $\\mathbb{E}\\left[\\mathrm{fix}_{\\gamma}\\left(N\\right)\\right]$ and determine the limit distribution of the number of fixed points as $N\\to\\infty$. These results are then generalized to all statistics of cycles of fixed lengths.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-23T07:49:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B15"
    ],
    "keywords": [
      "random permutation",
      "free product",
      "orientable hyperbolic surface group",
      "study local statistics",
      "fixed points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}